On Linear Inference Frank Pfennnig Summary: On Linear Inference Frank Pfennnig Draft of February 2, 2008 Inference. When we write an inference rule even(x) true even(s(s(x))) true what do we mean? To discuss this, some terminology: we say "even(t)" is a proposition and "even(t) true" is a judgment. Following Martin-L¨of, a judgment is an object of knowledge. We obtain knowledge by making inferences from judgments we already know. We can then read the rule above as If we know that t is even for a term t, we may conclude (and thereby know) that s(s(t)) is also even. The process of inference is therefore one by which we gain knowledge. We may start with the knowledge that 0 is even, then we gain the information that s(s(0)) is even, and next we learn that s(s(s(s(0)))) is even, and so on. This process is clearly monotonic: we gain knowledge, but we never forget, at least in the idealized world of mathematics. Persistent and Ephemeral Truth. It has long been argued by philosophers that truth is not universal, but depends on the world in which we consider a proposition. This could depend on time ("It is raining now, but it did not rain